The cheap answer (cheap because there should be a better one, but I don't know it) is that <$handle> seems to work, but <$array_of_handles[0]> doesn't. Changing 
my $this = <$files[$num]>;

[download]
to 
my $handle = $files[$num];
my $this = <$handle>;

[download]
should work.

By the way, you read one line too many. Your while test should be $totalLines < $wantedLines.

UPDATE: Thanks to Tanktalus for pointing me to the documentation. I/O Operators has this to say:

If what's within the angle brackets is neither a filehandle nor a simple scalar variable containing a filehandle name, typeglob, or typeglob reference, it is interpreted as a filename pattern to be globbed, and either a list of filenames or the next filename in the list is returned, depending on context. This distinction is determined on syntactic grounds alone. That means <$x> is always a readline() from an indirect handle, but <$hash{key}> is always a glob(). That's because $x is a simple scalar variable, but $hash{key} is not--it's a hash element. Even <$x > (note the extra space) is treated as glob("$x "), not readline($x).
As the documentation suggests, and almut and Perlbotics point out below, readline will probably serve better.

In reply to Re: reading from an arbitrary number of files using anonymous file handles by JadeNB
in thread reading from an arbitrary number of files using anonymous file handles by downer

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